Structure Estimation in Graphical Modelssteffen/seminars/waldstructure.pdf · Structure estimation...

IntroductionScore-based methods

Bayesian analysisConstraint-based methods

Summary and challengesThings I did not even get near

References

Structure Estimation in Graphical Models

Steffen Lauritzen, University of Oxford

Wald Lecture, World Meeting on Probability and Statistics

Istanbul 2012

Steffen Lauritzen, University of Oxford Structure Estimation in Graphical Models




References

Structure estimationSome examplesGeneral points

Advances in computing has set focus on estimation of structure:

I Model selection (e.g. subset selection in regression)

I System identification (engineering)

I Structural learning (AI or machine learning)





References


Advances in computing has set focus on estimation of structure:

I Model selection (e.g. subset selection in regression)

I System identification (engineering)

I Structural learning (AI or machine learning)

Graphical models describe conditional independence structures, sogood case for formal analysis.

Methods must scale well with data size, as many structures andhuge collections of data are to be considered.





References


Why estimation of structure?

I Parallel to e.g. density estimation

I Obtain quick overview of relations between variables incomplex systems

I Data mining

I Gene regulatory networks

I Reconstructing family trees from DNA information

I General interest in sparsity.





References


Markov mesh model





References


PC algorithm

Crudest algorithm (HUGIN), 10000 simulated cases





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Bayesian GES

Crudest algorithm (WinMine), 10000 simulated cases





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Tree model

PC algorithm, 10000 cases, correct reconstruction





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Bayesian GES on tree





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Chest clinic





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PC algorithm

10000 simulated cases





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Bayesian GES





References


SNPs and gene expressions

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min BIC forest





References


Methods for structure identification in graphical models can beclassified into three types:

I score-based methods: For example optimizing a penalizedlikelihood by using convex programming e.g. glasso;

I Bayesian methods: Identifying posterior distributions overgraphs; can also use posterior probability as score.

I constraint-based methods: Querying conditionalindependences and identifying compatible independencestructures, for example PC, PC*, NPC, IC, CI, FCI, SIN, QP,. . .





References

Estimating trees and forestsGraphical lassoScore-based methods for DAGs

Assume P factorizes w.r.t. an unknown tree τ . Based on a samplex (n) = (x1, . . . , xn) the likelihood function becomes

`(τ, p) = log L(τ) = log p(x (n) | τ)

=∑

e∈E(τ)

log p(x(n)e )−

∑v∈V{deg(v)− 1} log p(x

(n)v ).

Maximizing this over p for a fixed tree τ yields the profile likelihood

l(τ) = l(τ, p) =∑

e∈E(τ)

Hn(e) +∑v∈V

Hn(v)





References


Here Hn(e) is the empirical cross-entropy or mutual informationbetween endpoint variables of the edge e = {u, v}:

Hn(e) =∑ n(xu, xv )

nlog

n(xu, xv )/n

n(xu)n(xv )/n2

and similarly Hn(v) is the empirical entropy of Xv .

Based on this fact, (Chow and Liu, 1968) showed MLE τ of T hasmaximal weight, where the weight of τ is

w(τ) =∑

e∈E(τ)

Hn(e) = l(τ)− l(φ0).

Here φ0 is the graph with all vertices isolated.





References


Fast algorithms (Kruskal Jr., 1956) compute maximal weightspanning tree from weights W = (wuv , u, v ∈ V ).

Chow and Wagner (1978) show a.s. consistency in total variationof P: If P factorises w.r.t. τ , then

supx|p(x)− p(x)| → 0 for n→∞,

so if τ is unique for P, τ = τ for all n > N for some N.

If P does not factorize w.r.t. a tree, P converges to closesttree-approximation P to P (Kullback-Leibler distance).

Note that if P is Markov w.r.t. and undirected graph G and wedefine weights on edges as w(e) = H(e) by cross-entropy, P isMarkov w.r.t any maximal weight spanning tree of G.





References


Forests

Note that if we consider forests instead of trees, i.e. allow missingbranches, it is still true that

l(φ) = l(φ, p) =∑

e∈E(φ)

Hn(e) +∑v∈V

Hn(v).

Thus if we add a penalty for each edge, e.g. proportional to thenumber of additional parameters qe of introducing the edge e, wecan find the maximum penalised forest by Kruskal’s algorithmusing weights

w(φ) =∑

e∈E(φ)

{Hn(e)− λqe} .

This has been exploited in the package gRapHD (Edwards et al.,2010), see also Panayidou (2011) and Højsgaard et al. (2012).





References


SNPs and gene expressions

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min BIC forest





References


Gaussian Trees

If X = (Xv , v ∈ V ) is regular multivariate Gaussian, it factorizesw.r.t. an undirected graph if and only if its concentration matrixK = Σ−1 satisfies

kuv = 0 ⇐⇒ u 6∼ v .

Results of Chow et al. are easily extended to Gaussian trees, withthe weight of a tree determined as

w(τ) =∑

e∈E(τ)

− log(1− r2e ),

with r2e being the empirical correlation coeffient between Xu and

Xv .





References


I Direct likelihood methods (ignoring penalty terms) lead tosensible results.

I (Boootstrap) sampling distribution of tree MLE can besimulated

I Penalty terms additive along branches, so highest AIC or BICscoring tree (forest) also available using a MWST algorithm.

I Tree methods scale extremely well with both sample size andnumber of variables;

I Pairwise marginal counts are sufficient statistics for the treeproblem (empirical covariance matrix in the Gaussian case).





References


I Direct likelihood methods (ignoring penalty terms) lead tosensible results.

I (Boootstrap) sampling distribution of tree MLE can besimulated

I Penalty terms additive along branches, so highest AIC or BICscoring tree (forest) also available using a MWST algorithm.

I Tree methods scale extremely well with both sample size andnumber of variables;

I Pairwise marginal counts are sufficient statistics for the treeproblem (empirical covariance matrix in the Gaussian case).

Note sufficiency holds despite parameter space very different fromopen subset of Rk .





References


Consider an undirected Gaussian graphical model and thel1-penalized log-likelihood function

2`pen(K ) = log detK − tr(KW )− λ||K ||∗1.

The penalty ||K ||∗1 =∑

i 6=j |kij | is essentially a convex relaxation ofthe number of edges in the graph and optimization of thepenalized likelihood will typically lead to several kij = 0 and thus ineffect estimate a particular graph.

This penalized likelihood can be maximized efficiently (Banerjeeet al., 2008) as implemented in the graphical lasso (Friedmanet al., 2008).

Beware: not scale-invariant!





References


glasso for bodyfat

●Weight

●Knee

●Thigh

●Hip

●Abdomen

●Ankle ●Biceps

●Forearm

●Height

●Wrist

●Neck

●Chest

●Age

●BodyFat





References


Markov equivalence

D and D′ are equivalent if and only if:

1. D and D′ have same skeleton (ignoring directions)

2. D and D′ have same unmarried parents

so

s - ss@@R? s ≡ s - s s

s?@@I

but

s - s -

s@@R? s 6≡ s - s - s

s6@

@R





References


Equivalence class searches

Searches directly in equivalence classes of DAGs.

Define score function σ(D), with the property that

D ≡ D′ ⇒ σ(D) = σ(D′).

This holds e.g. if score function is AIC or BIC or full Bayesianposterior with strong hyper Markov prior (based upon Dirichlet orinverse Wishart distributions).

Equivalence class with maximal score is sought.

dlasso? problems with invariance over equivalence classes!





References


Greedy equivalence class search

1. Initialize with empty DAG

2. Repeatedly search among equivalence classes with a singleadditional edge and go to class with highest score - until noimprovement.

3. Repeatedly search among equivalence classes with a singleedge less and move to one with highest score - until noimprovement.

For BIC, for example, this algorithm yields consistent estimate ofequivalence class for P. (Chickering, 2002)





References

Basic Bayesian model estimationDecomposable graphical modelsBayesian methods for DAGs

For g in specified set of graphs, Θg is associated parameter spaceso that P factorizes w.r.t. g if and only if P = Pθ for some θ ∈ Θg .

πg is prior on Θg . Prior p(g) is uniform for simplicity.

Based on x (n), posterior distribution of G is

p∗(g) = p(g | x (n)) ∝ p(x (n) | g) =

∫Θg

p(x (n) | g , θ)πg (dθ).

Bayesian analysis looks for MAP estimate g∗ maximizing p∗(g) orattempts to sample from posterior, using e.g. MCMC.





References


For connected decomposable graphs and strong hyper Markovpriors Dawid and Lauritzen (1993) show

p(x (n) | g) =

∏C∈C p(x

(n)C )∏

S∈S p(x(n)S )

,

where each factor has explicit form. C are the cliques of g and Sthe separators (mininal cutsets).

Hence, if the prior distributions over a class of graphs is uniform,the posterior distribution has the form

p(g | x (n)) ∝∏

C∈C p(x(n)C )∏

S∈S p(x(n)S )

=

∏C∈C w(C )∏S∈S w(S)

.





References


Byrne (2011) shows that a distribution over decomposable graphshaving the form

p(g) ∝∏

C∈C w(C )∏S∈S w(S)

,

satisfies a structural Markov property so that for subsets A and Bwith V = A ∪ B, it holds that

GA⊥⊥GB | (A,B) is a decomposition of G.

In words, the finer structure of the graph bits in two componentsof any decomposition are independent.





References


Trees are decomposable, so for trees we get

p(τ | x (n)) ∝∏

e∈E(τ)

p(x(n)e )

which more illuminating can be expressed as

p(τ | x (n)) ∝∏

e∈E(τ)

Be

where

Buv =p(x

(n)uv )

p(x(n)u )p(x

(n)v )

is the Bayes factor for independence along the edge uv .





References


MAP estimates of trees can be computed (also in Gaussian case)

Good direct algorithms exist for generating random spanning trees(Guenoche, 1983; Broder, 1989; Aldous, 1990; Propp and Wilson,1998) so full posterior analysis is possible for trees.

MCMC methods for exploring posteriors of undirected graphs havebeen developed.

Even for forests, it seems complicated to sample from the posteriordistribution (Dai, 2008).

p(φ | x (n)) ∝∏

e∈E(φ)

Be





References


Some challenges for undirected graphs

I Find feasible algorithm for (perfect) simulation from adistribution over decomposable graphs as

p(g) ∝∏

C∈C w(C )∏S∈S w(S)

,

where w(A),A ⊆ V are a prescribed set of positive weights.

I Find feasible algorithm for obtaining MAP in decomposablecase. This may not be universally possible as problem isNP-complete, even for bounded maximum clique size.





References


Posterior distribution for DAG

For strong directed hyper Markov priors it holds that

p(x (n) | d) =∏v∈V

p(x(n)v | x (n)

pa(v))

sop(d | x (n)) ∝

∏v∈V

p(x(n)v | x (n)

pa(v)),

see e.g. Spiegelhalter and Lauritzen (1990); Cooper and Herskovits(1992); Heckerman et al. (1995)Challenge: Find good algorithm for sampling from this fullposterior.





References

PC algorithm

First step of constraint-based methods (eg PC-algorithm) is toidentify skeleton of G, which is the undirected graph with α 6∼ β ifand only if there exists S ⊆ V \ {α, β} with α⊥g β |S .

The skeleton stage of any such algorithm is thus correct for anytype of graph which represents the independence structure!

In particular, the the PC algorithm can easily be adapted to UGsand CHGs (Meek, 1996).

Challenge: the properties of these algorithms when translated tosampling situations are not sufficiently well understood.





References

PC algorithm

Step 1: Identify skeleton using that, for P faithful,

u 6∼ v ⇐⇒ ∃S ⊆ V \ {u, v} : Xu ⊥⊥Xv | XS .

Begin with complete graph, check for S = ∅ andremove edges when independence holds. Thencontinue for increasing |S |.PC-algorithm (Spirtes et al., 1993) exploits that onlyS with S ⊆ ne(u) or S ⊆ ne(v) needs checking, nerefers to current skeleton.

Step 2: Identify directions to be consistent withindependence relations found in Step 1.





References

PC algorithm

pcalg for bodyfat

BodyFat Age

Weight

Height

Neck

Chest

Abdomen

Hip

Thigh

Knee

Ankle

Biceps

Forearm

Wrist





References

PC algorithm

pcalg for carcass

Fat11

Meat11

Fat12

Meat12

Fat13

Meat13

LeanMeat





References

PC algorithm

Faithfulness

A given distribution P is in general compatible with a variety ofstructures, i.e. if P corresponds to complete independence. Toidentify a structure G something like the following must hold

P is said to be faithful to G if

A⊥⊥B | XS ⇐⇒ A⊥g B | S .

Most distributions are faithful. More precisely, for DAGs it holdsthat the non-faithful distributions form a Lebesgue null-set inparameter space associated with a DAG.





References

PC algorithm

Exact properties of PC-algorithm

If P is faithful to DAG D, PC-algorithm finds D′ equivalent to D.It uses N independence checks where N is at most

N ≤ 2

(|V |2

) d∑i=0

(|V | − 1

i

)≤ |V |

d+1

(d − 1)!,

where d is the maximal degree of any vertex in D. f So worst casecomplexity is exponential, but algorithm fast for sparse graphs.Sampling properties are less well understood although consistencyresults exist.





References

PC algorithm

Constraint based methods establish fundamentally two lists:

1. An independence list I of triplets (α, β, S) with α⊥⊥β |S ;identifies skeleton;

2. A dependence list D of triplets (α, β, S) with ¬(α⊥⊥β | S).

They get established recursively, for increasing size of S , and thelist I is most likely to have errors. The lists may or may not beconsistent with a DAG model, but methods exist for checking this.Question: Assume a current pair of input lists is consistent withcompositional graphoid axioms and a new triplet (α, β, S) isconsidered. Can it be verified whether it can be consistently addedto any of the two lists? Graph representable? Compositional?





References

I Seems out of hand to extend Bayesian and score basedmethods to more general graphical models;

I Fully Bayesian methods do typically not scale well withnumber of variables;

I Constraint based methods have less clear formal inferencebasis; challenge to improve this.

I Constraint based methods have been developed to work incases where P is faithful and conditional independence queriescan be resolved without error.





References

I Factorisation properties for Markov distributions;

I Local computation algorithms for speeding up any relevantcomputation;

I Causality and causal interpretations;

I Non-parametric graphical models for large scale data analysis;

I Probabilistic expert systems, for example in forensicidentification problems;

I Markov theory for infinite graphs (huge graphs).

I THANK YOU FOR LISTENING!





References

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Byrne, S. (2011). Hyper and Structural Markov Laws for GraphicalModels. PhD thesis, Statistical Laboratory, University ofCambridge.

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References

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References

Spirtes, P., Glymour, C., and Scheines, R. (1993). Causation,Prediction and Search. Springer-Verlag, New York. Reprinted byMIT Press.


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